Binding effects in high-energy scattering applied toK-shell ionization

Abstract

The equation describing scattering of a fast ion by a $K$-shell electron, $\frac{i\hslash v\partial S_I}{\partial Z}=\exp \left(\frac{i{\tilde{H}}_{e}Z}{\hslash v}\right)V(\overrightarrow{\mathrm{r}}, \overrightarrow{\mathrm{R}})\exp (-\frac{i{\tilde{H}}_{e}Z}{\hslash v})S_I$, where ${\tilde{H}}_e\left(\mathcal{r}\right)=-\left(\frac{{\hslash }^2}{2m_e}\right){\nabla }_{\mathcal{r}}^2-\left(\frac{{\hslash }^2}{m_e}\right)\left[\frac{\partial \ln {\chi }_0\left(\mathcal{r}\right)}{\partial \mathcal{r}}\right]\frac{\partial }{\partial \mathcal{r}}\equiv \mathrm{FT}+\mathrm{BT}$, is solved in the Glauber approximation by setting ${\tilde{H}}_e$ to zero and in the Cheshire approximation by setting the binding term (BT) to zero. In this paper we solve for $S_I$ including both the freely recoiling term (FT) and BT but neglecting their commutator. A divergence found with this method in a previous investigation of hydrogenlike atoms is removed as long as $\frac{Z_S{e}^2}{\hslash v}<1\mathrm{}$. This limit represents a natural "threshold" in the method since for lower velocities the target electron is moving faster than the projectile. We apply the method in its lowest-order approximation to $K$-shell ionization by differently charged projectiles. A substantial improvement in the fit to the ratio ${\mathcal{r}}_{12}$ given by ${\mathcal{r}}_{12}=\frac{\sigma \left(Z_2\right)Z_2^2}{\sigma \left(Z_2\right)Z_1^2}$, $Z_1>Z_2$, is found in comparison with the Glauber and Cheshire results. More experiments are needed with $Z_S\lesssim 20$ and at energies such that $\frac{Z_S{e}^2}{\hslash v}<1\mathrm{}$ to test this new theory. The method has immediate applicability to any scattering problem in which the projectile has a classical trajectory.